Editing Ideal gas law (section)

==Equation==

[[File:Ideal Gas Law.jpg|thumb|Molecular collisions within a closed container (a propane tank) are shown (right). The arrows represent the random motions and collisions of these molecules. The pressure and temperature of the gas are directly proportional: As temperature increases, the pressure of the propane gas increases by the same factor. A simple consequence of this proportionality is that on a hot summer day, the propane tank pressure will be elevated, and thus propane tanks must be rated to withstand such increases in pressure.]]

The [[state function|state]] of an amount of [[gas]] is determined by its pressure, volume, and temperature. The modern form of the equation relates these simply in two main forms. The temperature used in the equation of state is an absolute temperature: the appropriate [[SI unit]] is the [[kelvin]].<ref name="Equation of State">{{Cite web|url=http://www.grc.nasa.gov/WWW/K-12/airplane/eqstat.html|title=Equation of State|access-date=2010-08-29|archive-url=https://web.archive.org/web/20140823003747/http://www.grc.nasa.gov/WWW/k-12/airplane/eqstat.html|archive-date=2014-08-23|url-status=dead}}</ref>

===Common forms===

The most frequently introduced forms are:<math display="block">pV = nRT = n k_\text{B} N_\text{A} T = N k_\text{B} T </math>where:
* <math>p</math> is the absolute [[pressure]] of the gas,
* <math>V</math> is the [[volume]] of the gas,
* <math>n</math> is the [[amount of substance]] of gas (also known as number of moles),
* <math>R</math> is the ideal, or universal, [[gas constant]], equal to the product of the [[Boltzmann constant]] and the [[Avogadro constant]],
* <math>k_\text{B}</math> is the [[Boltzmann constant]],
* '''''<math>N_{A}</math>''''' is the [[Avogadro constant]],
* <math>T</math> is the [[Thermodynamic temperature|absolute temperature]] of the gas,
* <math>N</math> is the number of particles (usually atoms or molecules) of the gas.

In [[SI units]], ''p'' is measured in [[Pascal (unit)|pascals]], ''V'' is measured in [[cubic metre]]s, ''n'' is measured in [[Mole (unit)|moles]], and ''T'' in [[Kelvin (unit)|kelvins]] (the [[William Thomson, 1st Baron Kelvin|Kelvin]] scale is a shifted [[Celsius|Celsius scale]], where 0 K = −273.15&nbsp;°C, the [[Absolute zero|lowest possible temperature]]). ''R'' has for value 8.314 [[joule|J]]/([[Mole (unit)|mol]]·[[Kelvin|K]]) = 1.989 ≈ 2 [[calorie|cal]]/(mol·K), or 0.0821 L⋅[[Atmosphere (unit)|atm]]/(mol⋅K).

===Molar form===
How much gas is present could be specified by giving the mass instead of the chemical amount of gas. Therefore, an alternative form of the ideal gas law may be useful. The chemical amount, ''n'' (in moles), is equal to total mass of the gas (''m'') (in kilograms) divided by the [[molar mass]], ''M'' (in kilograms per mole):
: <math> n = \frac{m}{M}. </math>
By replacing ''n'' with ''m''/''M'' and subsequently introducing [[density]] ''ρ'' = ''m''/''V'', we get:
: <math> pV = \frac{m}{M} RT </math>
: <math> p = \frac{m}{V} \frac{RT}{M} </math>
: <math> p = \rho \frac{R}{M} T </math>
Defining the [[Gas constant#Specific gas constant|specific gas constant]] ''R''<sub>specific</sub> as the ratio ''R''/''M'',
: <math> p = \rho R_\text{specific}T </math>
This form of the ideal gas law is very useful because it links pressure, density, and temperature in a unique formula independent of the quantity of the considered gas.  Alternatively, the law may be written in terms of the [[specific volume]] ''v'', the reciprocal of density, as
: <math> pv = R_\text{specific}T. </math>

It is common, especially in engineering and meteorological applications, to represent the '''specific''' gas constant by the symbol ''R''. In such cases, the '''universal''' gas constant is usually given a different symbol such as <math>\bar R</math> or <math>R^*</math> to distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whether the universal or specific gas constant is being used.<ref>{{cite book |last1=Moran |last2=Shapiro |title=Fundamentals of Engineering Thermodynamics |publisher=Wiley |edition=4th |year=2000 |isbn=0-471-31713-6 }}</ref>

=== Statistical mechanics ===
In [[statistical mechanics]], the following molecular equation (i.e. the ideal gas law in its theoretical form) is derived from first principles:
: <math> p = nk_\text{B}T, </math>
where {{math|''p''}} is the absolute [[pressure]] of the gas, {{math|''n''}} is the [[number density]] of the molecules (given by the ratio {{math|1=''n'' = ''N''/''V''}}, in contrast to the previous formulation in which {{math|''n''}} is the ''number of moles''), {{math|''T''}} is the [[absolute temperature]], and {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]] relating temperature and energy, given by:
: <math> k_\text{B} = \frac{R}{N_\text{A}} </math>
where {{math|''N''<sub>A</sub>}} is the [[Avogadro constant]].
The form can be furtherly simplified by defining the kinetic energy corresponding to the temperature:
: <math> T := k_\text{B}T, </math>
so the ideal gas law is more simply expressed as:
: <math> p = n \, T, </math>

From this we notice that for a gas of mass {{math|''m''}}, with an average particle mass of {{math|''μ''}} times the [[atomic mass constant]], {{math|''m''<sub>u</sub>}}, (i.e., the mass is {{math|''μ''}}&nbsp;[[Dalton (unit)|Da]]) the number of molecules will be given by
: <math> N = \frac{m}{\mu m_\text{u}}, </math>
and since {{math|1=''ρ'' = ''m''/''V'' = ''nμm''<sub>u</sub>}}, we find that the ideal gas law can be rewritten as
: <math> p = \frac{1}{V}\frac{m}{\mu m_\text{u}} k_\text{B} T = \frac{k_\text{B}}{\mu m_\text{u}} \rho T. </math>

In SI units, {{math|''p''}} is measured in [[Pascal (unit)|pascals]], {{math|''V''}} in cubic metres, {{math|''T''}} in kelvins, and {{math|1=''k''<sub>B</sub> = {{physconst|k|ref=no|round=2}}}} in [[SI unit]]s.

=== Combined gas law ===
Combining the laws of Charles, Boyle, and Gay-Lussac gives the '''combined gas law''', which can take the same functional form as the ideal gas law. This form does not specify the number of moles, and the ratio of <math>PV</math> to <math>T</math> is simply taken as a constant:<ref>{{cite book |last1=Raymond |first1=Kenneth W. |title=General, organic, and biological chemistry : an integrated approach |publisher=John Wiley & Sons |isbn=9780470504765 |page=186 |edition=3rd |year=2010|url=https://books.google.com/books?id=iIltMoHUtJUC&pg=PA186 |access-date=29 January 2019}}</ref>
: <math>\frac{PV}{T}=k,</math>
where <math>P</math> is the [[pressure]] of the gas, <math>V</math> is the [[volume]] of the gas, <math>T</math> is the [[Thermodynamic temperature|absolute temperature]] of the gas, and <math>k</math> is a constant. More commonly, when comparing the same substance under two different sets of conditions, the law is written as:
: <math> \frac{P_1 V_1}{T_1}= \frac{P_2 V_2}{T_2}. </math>